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Mathematics - Circle Question with Solution | TestHub

MathematicsCircleCommon tangents to two circlesMedium2 minQB

MathematicsMediumsingle choice

Passage / Comprehension

The curve y exp(-βx²)=exp(α) passes through (1,1) with tangent slope -2. Let S₁: x²+y²-2αx-2βy+α²+β²-3=0 and S₂: x²+y²-12x-22y+130=0.

$y e^{- β x^{2}} = e^{α}$ passes through $(1, 1)$ with tangent slope -2 . Let $S_{1} : x^{2} + y^{2} - 2 αx - 2 β y + α^{2} + β^{2} - 3 = 0$ , and $S_{2} : x^{2} + y^{2} - 12 x - 22 y + 130 = 0$ . Find the shortest common tangent segment from $S_{1}$ to $S_{2}$ .

Options:

Answer:

Solution:

The centres are $(1, - 1), (6, 11)$ , separation 13. The radii are $3$ and $33$ because $36 + 121 - 130 = 27$ . The shortest (transverse) tangent segment has length $1 3^{2} - (43)^{2} = 11$ .

Stream:JEESubject:MathematicsTopic:CircleSubtopic:Common tangents to two circles

⏱ 2mℹ️ Source: QB

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